Next, to test whether the black hole can be destroyed, we consider a process where some matter falls into the black hole from a finite portion of the horizon and finally, the spacetime settles down to a static state, i.e. we assume that the black hole satisfies the stability condition [1]}, [2]}. This means, at a very early time, the spacetime geometry is the same as the background spacetime, i.e. there exists a coordinate system \(O_1: \lbrace v_1, r_1, \theta _1\rbrace \)  such that the metric has the same form with Eq. (REF ), i.e.,
\(ds^2=-f(r_1)dv^2_1+2\chi (r_1) dv_1 dr_1+r^2_1 h_{ij}d\theta ^i_1 d\theta ^j_1\,.\) 
